Set Packing Partitioning and Covering
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Rounding Algorithms for Covering Problems
Mathematical Programming 80 (1998) 63 89 Rounding algorithms for covering problems Dimitris Bertsimas a,,,1, Rakesh Vohra b,2 a Massachusetts Institute of Technology, Sloan School of Management, 50 Memorial Drive, Cambridge, MA 02142-1347, USA b Department of Management Science, Ohio State University, Ohio, USA Received 1 February 1994; received in revised form 1 January 1996 Abstract In the last 25 years approximation algorithms for discrete optimization problems have been in the center of research in the fields of mathematical programming and computer science. Re- cent results from computer science have identified barriers to the degree of approximability of discrete optimization problems unless P -- NP. As a result, as far as negative results are con- cerned a unifying picture is emerging. On the other hand, as far as particular approximation algorithms for different problems are concerned, the picture is not very clear. Different algo- rithms work for different problems and the insights gained from a successful analysis of a par- ticular problem rarely transfer to another. Our goal in this paper is to present a framework for the approximation of a class of integer programming problems (covering problems) through generic heuristics all based on rounding (deterministic using primal and dual information or randomized but with nonlinear rounding functions) of the optimal solution of a linear programming (LP) relaxation. We apply these generic heuristics to obtain in a systematic way many known as well as new results for the set covering, facility location, general covering, network design and cut covering problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. -
Selective Maximum Coverage and Set Packing
Selective Maximum Coverage and Set Packing Felix J. L. Willamowski1∗ and Bj¨ornF. Tauer2y 1 Lehrstuhl f¨urOperations Research RWTH Aachen University [email protected] 2 Lehrstuhl f¨urManagement Science RWTH Aachen University [email protected] Abstract. In this paper we introduce the selective maximum coverage and the selective maximum set packing problem and variants of them. Both problems are strongly related to well studied problems such as maximum coverage, set packing, and (bipartite) hypergraph matching. The two problems are given by a collection of subsets of a ground set and index subsets of the indices of these subsets. Additionally, there are weights either for each element of the ground set or each subset for each index subset. The goal is to find at most one index per index subset such that the total weight of covered elements or of disjoint subsets is maximum. Applications arise in transportation, e.g., dispatching for ride- sharing services. We prove strong intractability results for the problems and provide almost best possible approximation guarantees. 1 Introduction and Preliminaries We introduce the selective maximum coverage problem (smc) generalizing the weighted maximum k coverage problem (wmkc) which is given by a finite ground set X with weights w : X ! Q≥0, a finite collection S of subsets of X, and an 0 integer k 2 Z≥0. The goal is to find a subcollection S ⊆ S containing at most k P subsets and maximizing the total weight of covered elements x2X0 w(x) with 0 X = [S2S0 S. For unit weights, w ≡ 1, the problem is called maximum k coverage problem (mkc). -
Structural Graph Theory Meets Algorithms: Covering And
Structural Graph Theory Meets Algorithms: Covering and Connectivity Problems in Graphs Saeed Akhoondian Amiri Fakult¨atIV { Elektrotechnik und Informatik der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Rolf Niedermeier Gutachter: Prof. Dr. Stephan Kreutzer Gutachter: Prof. Dr. Marcin Pilipczuk Gutachter: Prof. Dr. Dimitrios Thilikos Tag der wissenschaftlichen Aussprache: 13. October 2017 Berlin 2017 2 This thesis is dedicated to my family, especially to my beautiful wife Atefe and my lovely son Shervin. 3 Contents Abstract iii Acknowledgementsv I. Introduction and Preliminaries1 1. Introduction2 1.0.1. General Techniques and Models......................3 1.1. Covering Problems.................................6 1.1.1. Covering Problems in Distributed Models: Case of Dominating Sets.6 1.1.2. Covering Problems in Directed Graphs: Finding Similar Patterns, the Case of Erd}os-P´osaproperty.......................9 1.2. Routing Problems in Directed Graphs...................... 11 1.2.1. Routing Problems............................. 11 1.2.2. Rerouting Problems............................ 12 1.3. Structure of the Thesis and Declaration of Authorship............. 14 2. Preliminaries and Notations 16 2.1. Basic Notations and Defnitions.......................... 16 2.1.1. Sets..................................... 16 2.1.2. Graphs................................... 16 2.2. Complexity Classes................................ -
3.1 Matchings and Factors: Matchings and Covers
1 3.1 Matchings and Factors: Matchings and Covers This copyrighted material is taken from Introduction to Graph Theory, 2nd Ed., by Doug West; and is not for further distribution beyond this course. These slides will be stored in a limited-access location on an IIT server and are not for distribution or use beyond Math 454/553. 2 Matchings 3.1.1 Definition A matching in a graph G is a set of non-loop edges with no shared endpoints. The vertices incident to the edges of a matching M are saturated by M (M-saturated); the others are unsaturated (M-unsaturated). A perfect matching in a graph is a matching that saturates every vertex. perfect matching M-unsaturated M-saturated M Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. 3 Perfect Matchings in Complete Bipartite Graphs a 1 The perfect matchings in a complete b 2 X,Y-bigraph with |X|=|Y| exactly c 3 correspond to the bijections d 4 f: X -> Y e 5 Therefore Kn,n has n! perfect f 6 matchings. g 7 Kn,n The complete graph Kn has a perfect matching iff… Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. 4 Perfect Matchings in Complete Graphs The complete graph Kn has a perfect matching iff n is even. So instead of Kn consider K2n. We count the perfect matchings in K2n by: (1) Selecting a vertex v (e.g., with the highest label) one choice u v (2) Selecting a vertex u to match to v K2n-2 2n-1 choices (3) Selecting a perfect matching on the rest of the vertices. -
Exact Algorithms and APX-Hardness Results for Geometric Packing and Covering Problems∗
Exact Algorithms and APX-Hardness Results for Geometric Packing and Covering Problems∗ Timothy M. Chan† Elyot Grant‡ March 29, 2012 Abstract We study several geometric set cover and set packing problems involv- ing configurations of points and geometric objects in Euclidean space. We show that it is APX-hard to compute a minimum cover of a set of points in the plane by a family of axis-aligned fat rectangles, even when each rectangle is an ǫ-perturbed copy of a single unit square. We extend this result to several other classes of objects including almost-circular ellipses, axis-aligned slabs, downward shadows of line segments, downward shad- ows of graphs of cubic functions, fat semi-infinite wedges, 3-dimensional unit balls, and axis-aligned cubes, as well as some related hitting set prob- lems. We also prove the APX-hardness of a related family of discrete set packing problems. Our hardness results are all proven by encoding a highly structured minimum vertex cover problem which we believe may be of independent interest. In contrast, we give a polynomial-time dynamic programming algo- rithm for geometric set cover where the objects are pseudodisks containing the origin or are downward shadows of pairwise 2-intersecting x-monotone curves. Our algorithm extends to the weighted case where a minimum-cost cover is required. We give similar algorithms for several related hitting set and discrete packing problems. 1 Introduction In a geometric set cover problem, we are given a range space (X, )—a universe X of points in Euclidean space and a pre-specified configurationS of regions or geometric objects such as rectangles or half-planes. -
Bin Completion Algorithms for Multicontainer Packing, Knapsack, and Covering Problems
Journal of Artificial Intelligence Research 28 (2007) 393-429 Submitted 6/06; published 3/07 Bin Completion Algorithms for Multicontainer Packing, Knapsack, and Covering Problems Alex S. Fukunaga [email protected] Jet Propulsion Laboratory California Institute of Technology 4800 Oak Grove Drive Pasadena, CA 91108 USA Richard E. Korf [email protected] Computer Science Department University of California, Los Angeles Los Angeles, CA 90095 Abstract Many combinatorial optimization problems such as the bin packing and multiple knap- sack problems involve assigning a set of discrete objects to multiple containers. These prob- lems can be used to model task and resource allocation problems in multi-agent systems and distributed systms, and can also be found as subproblems of scheduling problems. We propose bin completion, a branch-and-bound strategy for one-dimensional, multicontainer packing problems. Bin completion combines a bin-oriented search space with a powerful dominance criterion that enables us to prune much of the space. The performance of the basic bin completion framework can be enhanced by using a number of extensions, in- cluding nogood-based pruning techniques that allow further exploitation of the dominance criterion. Bin completion is applied to four problems: multiple knapsack, bin covering, min-cost covering, and bin packing. We show that our bin completion algorithms yield new, state-of-the-art results for the multiple knapsack, bin covering, and min-cost cov- ering problems, outperforming previous algorithms by several orders of magnitude with respect to runtime on some classes of hard, random problem instances. For the bin pack- ing problem, we demonstrate significant improvements compared to most previous results, but show that bin completion is not competitive with current state-of-the-art cutting-stock based approaches. -
Solving Packing Problems with Few Small Items Using Rainbow Matchings
Solving Packing Problems with Few Small Items Using Rainbow Matchings Max Bannach Institute for Theoretical Computer Science, Universität zu Lübeck, Lübeck, Germany [email protected] Sebastian Berndt Institute for IT Security, Universität zu Lübeck, Lübeck, Germany [email protected] Marten Maack Department of Computer Science, Universität Kiel, Kiel, Germany [email protected] Matthias Mnich Institut für Algorithmen und Komplexität, TU Hamburg, Hamburg, Germany [email protected] Alexandra Lassota Department of Computer Science, Universität Kiel, Kiel, Germany [email protected] Malin Rau Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, LIG, 38000 Grenoble, France [email protected] Malte Skambath Department of Computer Science, Universität Kiel, Kiel, Germany [email protected] Abstract An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no “small” items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number k of small items. Our main results are fixed-parameter algorithms for vector versions of Bin Packing, Multiple Knapsack, and Bin Covering parameterized by k. The algorithms are randomized with one-sided error and run in time 4k · k! · nO(1). To achieve this, we introduce a colored matching problem to which we reduce all these packing problems. The colored matching problem is natural in itself and we expect it to be useful for other applications. -
Reductions and Satisfiability
Reductions and Satisfiability 1 Polynomial-Time Reductions reformulating problems reformulating a problem in polynomial time independent set and vertex cover reducing vertex cover to set cover 2 The Satisfiability Problem satisfying truth assignments SAT and 3-SAT reducing 3-SAT to independent set transitivity of reductions CS 401/MCS 401 Lecture 18 Computer Algorithms I Jan Verschelde, 30 July 2018 Computer Algorithms I (CS 401/MCS 401) Reductions and Satifiability L-18 30 July 2018 1 / 45 Reductions and Satifiability 1 Polynomial-Time Reductions reformulating problems reformulating a problem in polynomial time independent set and vertex cover reducing vertex cover to set cover 2 The Satisfiability Problem satisfying truth assignments SAT and 3-SAT reducing 3-SAT to independent set transitivity of reductions Computer Algorithms I (CS 401/MCS 401) Reductions and Satifiability L-18 30 July 2018 2 / 45 reformulating problems The Ford-Fulkerson algorithm computes maximum flow. By reduction to a flow problem, we could solve the following problems: bipartite matching, circulation with demands, survey design, and airline scheduling. Because the Ford-Fulkerson is an efficient algorithm, all those problems can be solved efficiently as well. Our plan for the remainder of the course is to explore computationally hard problems. Computer Algorithms I (CS 401/MCS 401) Reductions and Satifiability L-18 30 July 2018 3 / 45 imagine a meeting with your boss ... From Computers and intractability. A Guide to the Theory of NP-Completeness by Michael R. Garey and David S. Johnson, Bell Laboratories, 1979. Computer Algorithms I (CS 401/MCS 401) Reductions and Satifiability L-18 30 July 2018 4 / 45 what you want to say is From Computers and intractability. -
8. NP and Computational Intractability
Chapter 8 NP and Computational Intractability CS 350 Winter 2018 1 Algorithm Design Patterns and Anti-Patterns Algorithm design patterns. Ex. Greedy. O(n log n) interval scheduling. Divide-and-conquer. O(n log n) FFT. Dynamic programming. O(n2) edit distance. Duality. O(n3) bipartite matching. Reductions. Local search. Randomization. Algorithm design anti-patterns. NP-completeness. O(nk) algorithm unlikely. PSPACE-completeness. O(nk) certification algorithm unlikely. Undecidability. No algorithm possible. 2 8.1 Polynomial-Time Reductions Classify Problems According to Computational Requirements Q. Which problems will we be able to solve in practice? A working definition. [von Neumann 1953, Godel 1956, Cobham 1964, Edmonds 1965, Rabin 1966] Those with polynomial-time algorithms. Yes Probably no Shortest path Longest path Matching 3D-matching Min cut Max cut 2-SAT 3-SAT Planar 4-color Planar 3-color Bipartite vertex cover Vertex cover Primality testing Factoring 4 Classify Problems Desiderata. Classify problems according to those that can be solved in polynomial-time and those that cannot. Provably requires exponential-time. Given a Turing machine, does it halt in at most k steps? (the Halting Problem) Given a board position in an n-by-n generalization of chess, can black guarantee a win? Frustrating news. Huge number of fundamental problems have defied classification for decades. This chapter. Show that these fundamental problems are "computationally equivalent" and appear to be different manifestations of one really hard problem. 5 Polynomial-Time Reduction Desiderata'. Suppose we could solve X in polynomial-time. What else could we solve in polynomial time? don't confuse with reduces from Reduction. -
An Overview of Graph Covering and Partitioning
Takustr. 7 Zuse Institute Berlin 14195 Berlin Germany STEPHAN SCHWARTZ An Overview of Graph Covering and Partitioning ZIB Report 20-24 (August 2020) Zuse Institute Berlin Takustr. 7 14195 Berlin Germany Telephone: +49 30-84185-0 Telefax: +49 30-84185-125 E-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 An Overview of Graph Covering and Partitioning Stephan Schwartz Abstract While graph covering is a fundamental and well-studied problem, this eld lacks a broad and unied literature review. The holistic overview of graph covering given in this article attempts to close this gap. The focus lies on a characterization and classication of the dierent problems discussed in the literature. In addition, notable results and common approaches are also included. Whenever appropriate, our review extends to the corresponding partioning problems. Graph covering problems are among the most classical and central subjects in graph theory. They also play a huge role in many mathematical models for various real-world applications. There are two dierent variants that are concerned with covering the edges and, respectively, the vertices of a graph. Both draw a lot of scientic attention and are subject to prolic research. In this paper we attempt to give an overview of the eld of graph covering problems. In a graph covering problem we are given a graph G and a set of possible subgraphs of G. Following the terminology of Knauer and Ueckerdt [KU16], we call G the host graph while the set of possible subgraphs forms the template class. -
Approximation Algorithm for Vertex Cover with Multiple Covering Constraints
Approximation Algorithm for Vertex Cover with Multiple Covering Constraints Eunpyeong Hong Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan [email protected] Mong-Jen Kao Department of Computer Science and Information Engineering, National Chung-Cheng University, Chiayi, Taiwan [email protected] Abstract We consider the vertex cover problem with multiple coverage constraints in hypergraphs. In this problem, we are given a hypergraph G = (V, E) with a maximum edge size f, a cost function + w : V → Z , and edge subsets P1,P2,...,Pr of E along with covering requirements k1, k2, . , kr for each subset. The objective is to find a minimum cost subset S of V such that, for each edge subset Pi, at least ki edges of it are covered by S. This problem is a basic yet general form of classical vertex cover problem and a generalization of the edge-partitioned vertex cover problem considered by Bera et al. We present a primal-dual algorithm yielding an (f · Hr + Hr)-approximation for this problem, th where Hr is the r harmonic number. This improves over the previous ratio of (3cf log r), where c is a large constant used to ensure a low failure probability for Monte-Carlo randomized algorithms. Compared to previous result, our algorithm is deterministic and pure combinatorial, meaning that no Ellipsoid solver is required for this basic problem. Our result can be seen as a novel reinterpretation of a few classical tight results using the language of LP primal-duality. 2012 ACM Subject Classification Mathematics of computing → Approximation algorithms Keywords and phrases Vertex cover, multiple cover constraints, Approximation algorithm Digital Object Identifier 10.4230/LIPIcs.ISAAC.2018.43 Funding This work is supported in part by Ministry of Science and Technology (MOST), Taiwan, under Grants MOST107-2218-E-194-015-MY3 and MOST106-2221-E-001-006-MY3. -
Graph Covering and Subgraph Problems
Graph Covering and Subgraph Problems Gabrovˇsek, Peter and Miheliˇc, Jurij Abstract: Combinatorial optimization prob- For each problem, we first present its for- lems and graph theory play an important role in mal definition and give its hardness of solving. several fields including logistics, artificial intel- Afterward, we also list several exact, heuristic ligence, software engineering, etc. In this pa- and approximation algorithms for solving it. per, we explore a subset of combinatorial graph problems which we placed into two groups, Our survey methodology is as follows. First namely, graph covering and subgraph problems. we investigated definitions in several generic 1 Here, the former include well-known dominat- sources such as online compendiums , encyclo- ing set and vertex cover problem, and the lat- pedias2 and books [1, 13, 14]. Afterward, we ter include clique and independent set prob- searched for the research papers on the focused lem. For each problem we give its definition, graph problems using Google scholar. Our goal report state-of-the-art exact algorithms as well was to supplement each problem with refer- as heuristic and approximation algorithms, and list real-world applications if we were able to ences to exact, heuristics and approximation obtain any. At the end of the paper, we also algorithms. Moreover, we also supplemented relate the problems among each other. our descriptions with real-world applications. Finally, we also collected a list of reductions Index Terms: combinatorial optimization, among the selected problems. graph covering, subgraph problems, exact al- Let us present a few basic notions used in gorithms, approximation algorithms the rest of the paper.